The component counts of random functions

Abstract

Consider functions f:A→A∪C, where A and C are disjoint finite sets. The weakly connected components of the digraph of such a function are cycles of rooted trees, as in random mappings, and isolated rooted trees. Let n1=|A| and n3=|C|. When a function is chosen from all (n1+n3)n1 possibilities uniformly at random, then we find the following limiting behaviour as n1→∞. If n3=o(n1), then the size of the maximal mapping component goes to infinity almost surely; if n3∼γn1, γ>0 a constant, then process counting numbers of mapping components of different sizes converges; if n1=o(n3), then the number of mapping components converges to 0 in probability. We get estimates on the size of the largest tree component which are of order log⁡n3 when n3∼γn1 and constant when n3∼n1α, α>1. These results are similar to ones obtained previously for random injections, for which the weakly connected components are cycles and linear trees.